Hedging Interest Rate Risk. Treasury/Eurodollar Futures. Derivative Securities. Stocks and Bonds represent claims to specific future cash flows Derivative securities on the other hand represent contracts that designate future transactions
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Treasury/Eurodollar Futures
A futures contract describes a transaction (Commodity, Price, and Quantity) that will be made in the future.
In “Trading Places” (1983), Eddie Murphy and Dan Ackroyd were trading Orange Juice Futures
Orange Juice futures (FCOJ) are traded on the NYBOT (New York Board of Trade)
Contract = 15,000 Lbs. ; Price = cents/lb
Every contract must have two participants (Long = Buy, Short = Sell)
Alongposition in MAR FCOJ would require you to purchase FCOJ in March
A short position in JUL FCOJ would require you to deliver FCOJ in March
Now
Mar
Apr
May
June
July
Aug
Dealers pass orders along to the pit traders who create a contract.
Long
Short
3 May Contracts (15k * 3 = 45k lbs.) @ 88 cents/lb.
The contracts are then passed along to the exchange who will become the middleman
Note: the exchange is holding two contracts with a zero net position
Short (3 contracts)
Long (3 contracts)
Long (3 Contracts)
Short (3 Contracts)
First Notice Date
Last Delivery Date
Last Trading Day
First Delivery Date
Last Notice Date
May 1
May 8
May 10
May 23
May 31
Suppose that, on May 3, the short position decides that he wants out of the contract. The current May futures price is .92 per Lb
He could take a long position on 3 May contracts at a price of .92/LB
3 Contracts (Short) @ .88/LB
This would effectively “cancel out” the previous position at a loss of 3 cents/LB
.03*45,000 = $1,350 Loss
May 1
May 8
May 10
May 23
May 31
Suppose that, on May 12, the short position opts for delivery of the commodity. The current spot price is .84 per Lb
3 Contracts (Short) @ .88/LB
The Exchange Pairs up Longs with Shorts
3 Contracts (Long) @ .88/LB
Profit = (.88.84)*45,000
= $1,800
Loss = (.88.84)*45,000
= $1,800
May 1
May 8
May 10
May 23
May 31
These contracts are settled on a cash basis:Short Position Profits = (F – S)*500
Long Position Profits = (S – F)*500
F = Futures Price, S = Current Spot Price
Profits from price increases
Long Position
Short Position
Profits from price decreases
Treasury futures first began trading on the CME in 1976. The underlying commodity is a Treasury Bill, Note, or Bond. Remember, when interest rates rise, Treasury prices fall!
Profits from price increases
Profits from decreasing interest rates
Long Position
Profits from price decreases
Profits from increasing interest rates
Short Position
With TBill Futures, the commodity is a $1M Treasury Bill with 3 months left until maturity
Contracts exist for February, March, April, June, September, and December delivery
Last Trading Day (TBill Auction)
First Trading Day
Delivery Day
Nov 16, 2004
Feb 14
Feb 18
We have already calculated the Yield to Maturity for 90 Day Treasury Bills
Face Value  Price
365
YTM =
*100
Price
t
Days left until maturity
Annualized
Often, the yield referred to for Treasury Bills is the discount yield
Annualized with a 360 day year
Face Value  Price
360
DY =
*100
Face Value
t
Interest As a percentage of Face Value rather than Price
TBill futures are listed using the IMM (International Monetary Market) Index
IMM = 100 – Discount Yield
For example, if the Price of a $100, 90 Day Treasury were $98.
$100  $98
360
DY =
*100 = 8%
$100
90
IMM = 100 – 8 = 92
Every .005 increase in the IMM raises the value of a long TBill position by $12.50 ($25 per basis point).
IMM = 100 – LIBOR
Every .005 increase in the IMM raises the value of the long position by $12.50. ($25 per basis point)
Suppose that a march Eurodollar future (expires in 47 days) was currently selling for 94.555
We also have the current money rates (LIBOR)
IMM = 100  LIBOR
This contract is paying an annualized (yield) of 100 – 94.555 = 5.445%
The Eurodollar Future currently has an annual yield of 5.445%
5.445
= 1.3613%
4
$1M (1.013613) = $1,013,613
Delivery of a $1M 90Day Eurodollar account
Purchase/Sale of Eurodollar Future
Receipt of $1,013,613
Now
Day 47
Day 137
90 Days
Use a linear interpolation to get the 47 day spot rate 5.445%
47
5.2175%
= .6811%
360
47 Day Return
Yield
5.3125%
5.2175%
5.18%
Term
1 Month
3 Months
47 Days
Use a linear interpolation to get the 137 day spot rate 5.445%
137
5.4855%
= 2.0875%
360
137 Day Return
Yield
5.6438%
5.4855%
5.3125%
Term
3 Months
6 Months
137 Days
The Eurodollar Future currently has an annual yield of 5.445%
5.445
= 1.3613%
4
S(47) = .6811%
S(137) = 2.0875%
Now
Day 47
Day 137
1.020875
=1.01397 = 1.3970% =
F(47,90)
1.006811
The Eurodollar Future currently has an annual yield of 5.445% (1.3613%)
IMM = 100 – 5.445 = 94.555
The implied noarbitrage interest rate between 47 and 137 days is 5.588% (1.3970%)
IMM = 100 – 5.588 = 94.412
The interest rate on the futures contract is to low!!
or, alternatively
The price of the futures contract is too high!!!
Borrow at Futures Rate (Sell a Futures contract)
Now
Day 47
Day 137
Profit =
1.013970 – 1.013613
$1M = $357
Lend at the implied forward rate
How do you lend at the implied forward rate? 5.445%
By lending for the entire 137 day period and borrowing for the first 47 days, your net position is as a lender for the last 90 day period!
Borrow
Lend
Now
Day 47
Day 137
Go Short on a the futures contract at a price of 94.555 5.445%
Lend $992,885 for 137 days at the spot rate of 5.4855% (You will be paid $1,013,613 in 137 days)
Borrow $992,885 for 47 days at the spot rate of 5.2175%
Receive $1,013,613 from the original 137 day loan
Pay $1,013,613 on the 90 day loan
Borrow $1,000,000 at the rate established by the futures contract (5.445%)
Pay back the $992,885 Loan + interest ($999,643)
Now
Day 47
Day 137
On the 47 5.445%th day, you get a net cash flow of $352. This is the present value of $357 dollars to be received in 90 Days (you get the profits on day 47 rather than day 137)
The no arbitrage price of a price of a futures contract will reflect the forward rate implied by the yield curve. But remember, the forward rate is the expected future spot rate
Futures Rate = Expected Future Spot Rate
The commodity for TNote/Bond futures is a Treasury with a 6% annual coupon. What if there are no 6% bonds available?
Treasury Note/Bond futures are based on cheapest to deliver (CTD)basis.
It’s the short position’s option to deliver whatever has the lowest cost
Suppose that you have a short position on a a Treasury bond future that expires this month (any bond with an expiration date between 2020 and 2030 would be acceptable for delivery:
The cheapest to deliver bond will always be the lowest coupon, longest maturity bond
The conversion factors are meant to make all deliverable bonds “equally attractive”
Invoice Amount
Contract Size
Futures Price
Conversion Factor
Accrued Interest
=
It’s the short position’s option to deliver whatever has the lowest cost
To Find the cheapest to deliver bond/note
Current Futures Price
Conversion Factor
Spot Price
Maximize

Note: This will always be negative
20 Year Treasury Delivered
20 Year Treasury Expires
Now
March
March 2025
The Logic behind pricing treasury note/bond futures is the same as with TBill futures. The price should reflect expectation of future spot rates. However, note that expectations of future spot rates are already incorporated in bond prices!
Expected Future Treasury Price
Futures Price =
+ (Carry Costs – Carry Return)
Arbitrage Costs
Lets return to the 5 year Treasury Note example. Interest rates are currently 5% and are expected to stay at 5% (the yield curve is flat). A 5 year treasury note with $500,000 of face value and a 5% annual coupon.
$25,000
$25,000
$25,000
$25,000
$525,000
P
+
+
+
+
=
= $500,000
2
3
4
5
(1.05)
(1.05)
(1.05)
(1.05)
(1.05)
We already calculated the Modified Duration for this bond
MD = 4.3
That is, a 100 basis point increase in the interest rate lowers this bond’s price by (.043)($500,000) = $21,500
Profits from price decreases
Profits from increasing interest rates
Short Position (Futures)
If you are long in bonds, you are worried about rising interest rates (rising interest rates lower the value of your bond). Therefore, you could hedge this risk by holding short positions in TBill futures (Short positions make money when interest rates drop)
Profits from price decreases
Profits from increasing interest rates
Short Position (Futures)
A perfect hedge eliminates all your interest rate risk
Change in value of each contract
Change in value of bond position
Change in value of Value of Futures position
# of Futures Contracts
=
=
$2,500
$21,500
$21,500/$2,500 = 8.6 Contracts
Change in value of each contract
Change in value of bond position
Change in value of Value of Futures position
# of Futures Contracts
=
=
$2,500
$21,500
$21,500/$2,500 = 8.6 Contracts
Dollar Duration of Bonds
MD(B)
FV(B)
4.3
$500K
=
Hedge Ratio =
=
MD(F)
FV(F)
.25
$1M
Dollar Duration of Futures
One Problem….. 6% annual coupon. What if there are no 6% bonds available?
X 100
Here we have the 5 year Treasury key durations. Note that this bond’s price is most sensitive to the 5 Year spot rate. The future’s value is based on the 90 day treasury rate
Change in value of each contract 6% annual coupon. What if there are no 6% bonds available?
Change in value of bond position
Change in value of Value of Futures position
# of Futures Contracts
=
=
$2,500
$21,500
$21,500/$2,500 = 8.6 Contracts
We assumed that the 90 Day TBill rate and the 5 Year Rate were perfectly correlated. Suppose, instead, that we have
Change in 90 Day Treasury Rate
Change in 5 Year Rate
= (.5)
The hedge ratio drops to 4.3!